nForum - Discussion Feed (parameterized cohesive spectra) 2023-12-09T21:33:10+00:00 https://nforum.ncatlab.org/ Lussumo Vanilla & Feed Publisher David_Corfield comments on "parameterized cohesive spectra" (42606) https://nforum.ncatlab.org/discussion/5321/?Focus=42606#Comment_42606 2013-10-24T19:02:13+00:00 2023-12-09T21:33:09+00:00 David_Corfield https://nforum.ncatlab.org/account/20/ Now I’m not sure what the ’it’ of indeed it’s not refers to. Or the ’this’ of I already said this. I was expecting H I\mathbf{H}^I not to be an infinitesimal thickening of ...

Now I’m not sure what the ’it’ of

indeed it’s not

refers to. Or the ’this’ of

I was expecting $\mathbf{H}^I$ not to be an infinitesimal thickening of $\mathbf{H}$. I was wondering if this fact could already be seen from the failure of $\infty Grpd^I$ to be infinitesimally cohesive over $\infty Grpd$.

It’s part of my transporting thickenings idea. Were $\mathbf{H}^I$ a thickening over $\mathbf{H}$ we could transport it to $\infty Grpd^I$ over $\infty Grpd$. But we know we can’t.

But maybe the transporting idea isn’t sound.

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Urs comments on "parameterized cohesive spectra" (42605) https://nforum.ncatlab.org/discussion/5321/?Focus=42605#Comment_42605 2013-10-24T18:35:19+00:00 2023-12-09T21:33:09+00:00 Urs https://nforum.ncatlab.org/account/4/ Yes, indeed it’s not. Sorry, I thought I already said this. In a way it is the finite length of the interval (0&rightarrow;1)(0 \to 1) that prevents it, for it forces the neighbouring adjoints ...

Yes, indeed it’s not. Sorry, I thought I already said this. In a way it is the finite length of the interval $(0 \to 1)$ that prevents it, for it forces the neighbouring adjoints to be different, which for $T \mathbf{H}$ coincide.

By the way, I have now added on p. 256 more remarks previewing the way that the homotopy cofiber of $\infty$-toposes along a differential cohesion inclusion produces the underlying infinitesimal cohesion, here https://dl.dropboxusercontent.com/u/12630719/cohesivedocument131024.pdf.

This is quite probably true in full generality, but right now I just show it in the examples.

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David_Corfield comments on "parameterized cohesive spectra" (42604) https://nforum.ncatlab.org/discussion/5321/?Focus=42604#Comment_42604 2013-10-24T18:15:06+00:00 2023-12-09T21:33:09+00:00 David_Corfield https://nforum.ncatlab.org/account/20/ I guess I was hoping that exponentiation by II is simple enough that the failure of &infin;Grpd I\infty Grpd^I to be infinitesimally cohesive would imply that H IH^I is not differentially ...

I guess I was hoping that exponentiation by $I$ is simple enough that the failure of $\infty Grpd^I$ to be infinitesimally cohesive would imply that $H^I$ is not differentially cohesive for cohesive $H$.

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Urs comments on "parameterized cohesive spectra" (42603) https://nforum.ncatlab.org/discussion/5321/?Focus=42603#Comment_42603 2013-10-24T17:45:16+00:00 2023-12-09T21:33:09+00:00 Urs https://nforum.ncatlab.org/account/4/ I need to think about those higher jet toposes more in order to answer, but meanwhile just one comment to clear up terminoloy (since I may have cause a mixup here) I am saying “infinitesimally ...

I need to think about those higher jet toposes more in order to answer, but meanwhile just one comment to clear up terminoloy (since I may have cause a mixup here)

I am saying

Now if $\mathbf{H}$ is infinitesimally cohesive over $\infty\mathrm{Grpd}$, then the inclusion $\infty Grpd \hookrightarrow \mathbf{H}$ also exhibits $\mathbf{H}$ as being differentially cohesive over $\infty Grpd$. But the converse does not in general hold.

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David_Corfield comments on "parameterized cohesive spectra" (42602) https://nforum.ncatlab.org/discussion/5321/?Focus=42602#Comment_42602 2013-10-24T17:18:57+00:00 2023-12-09T21:33:09+00:00 David_Corfield https://nforum.ncatlab.org/account/20/ Yes I knew that it wasn’t differentially cohesive as the modalities don’t equate. But the original question was then where in the tower of cohesive jet toposes does differential cohesiveness ...

Yes I knew that it wasn’t differentially cohesive as the modalities don’t equate. But the original question was then where in the tower of cohesive jet toposes does differential cohesiveness fail. If $J^1 = T$ is differential, does it already fail at $J^2$?

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Urs comments on "parameterized cohesive spectra" (42601) https://nforum.ncatlab.org/discussion/5321/?Focus=42601#Comment_42601 2013-10-24T17:05:41+00:00 2023-12-09T21:33:09+00:00 Urs https://nforum.ncatlab.org/account/4/ Oh, I see. Yes, my fault. Fixed now. Concerning differentially cohesive: ah, now I see what you mean. No, not over H\mathbf{H}, unless I am missing something. True. What makes one kind of cohesive ...

Oh, I see. Yes, my fault. Fixed now.

Concerning differentially cohesive: ah, now I see what you mean. No, not over $\mathbf{H}$, unless I am missing something. True.

What makes one kind of cohesive homtopy types $X$ be “differential cohesive” over another $x$ is if they are “infinitesimal thickenings” of that latter, meaning that you cannot see the thickening of $X$ when homming into it out of $x$; something non-trivial only happens the other way around, homming $X$ into $X$ sees “tangents” in $x$.

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David_Corfield comments on "parameterized cohesive spectra" (42600) https://nforum.ncatlab.org/discussion/5321/?Focus=42600#Comment_42600 2013-10-24T16:55:21+00:00 2023-12-09T21:33:09+00:00 David_Corfield https://nforum.ncatlab.org/account/20/ Google must be picking up similar misspellings then. It’s double c and double m: accommodate. (Latin: ad-commodus, probably in turn con-modus) Yes H I\mathbf{H}^I is cohesive, but is it ...

Google must be picking up similar misspellings then. It’s double c and double m: accommodate. (Latin: ad-commodus, probably in turn con-modus)

Yes $\mathbf{H}^I$ is cohesive, but is it differentially cohesive?

That would be great if you would look over especially the physics slides (when they’re ready).

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Urs comments on "parameterized cohesive spectra" (42598) https://nforum.ncatlab.org/discussion/5321/?Focus=42598#Comment_42598 2013-10-24T16:42:26+00:00 2023-12-09T21:33:09+00:00 Urs https://nforum.ncatlab.org/account/4/ Thanks! Concerning “accomodate”, isng’t that correct? I checked with Google and Google seems to agree? Did you maybe make a reverse typo when reporting a typo? Concerning H I\mathbf{H}^I: ...

Thanks!

Concerning “accomodate”, isng’t that correct? I checked with Google and Google seems to agree? Did you maybe make a reverse typo when reporting a typo?

Concerning $\mathbf{H}^I$: this is in fact cohesive over $\mathbf{H}$, but my section on it doesn’t mention that (yet). Back when I wrote that section I used $\mathbf{H}^I$ for other purposes than giving twisted cohomology an internal home. It is now only through the company of the flashy $T \mathbf{H}$ that its superficially boring cousin $\mathbf{H}^I$ is getting some recognition, too.

Concerning your talk: interesting! If you have any preliminary notes, I’d be happy to look through them and return a tiny bit of the favor.

BTW, that same week I will be speaking in Edinburgh on “Higher toposes of laws of motion”.

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David_Corfield comments on "parameterized cohesive spectra" (42597) https://nforum.ncatlab.org/discussion/5321/?Focus=42597#Comment_42597 2013-10-24T16:24:27+00:00 2023-12-09T21:33:09+00:00 David_Corfield https://nforum.ncatlab.org/account/20/ That made me see ’accomodate’ twice ’projetion’ Since THT \mathbf{H} is differentially cohesive over a cohesive H\mathbf{H}, but H &Delta;\mathbf{H}^{\Delta} is not, at what ...

’accomodate’ twice

’projetion’

Since $T \mathbf{H}$ is differentially cohesive over a cohesive $\mathbf{H}$, but $\mathbf{H}^{\Delta}$ is not, at what point in the $J^n(\mathbf{H})$ interpolating between them does this property go?

By the way, it’s not just altruism that has me reading your book. I’ve promised to give a talk with the grand title Homotopy Type Theory: a revolutionary language for philosophy of logic, mathematics, and physics?. I’m nervous about getting the physics part up to scratch. Think I might talk about covariance. Maybe also something on quantization.

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Urs comments on "parameterized cohesive spectra" (42595) https://nforum.ncatlab.org/discussion/5321/?Focus=42595#Comment_42595 2013-10-24T15:32:19+00:00 2023-12-09T21:33:09+00:00 Urs https://nforum.ncatlab.org/account/4/ I have now added some notes on this to section 4.1.2 of https://dl.dropboxusercontent.com/u/12630719/cohesivedocument131024.pdf

I have now added some notes on this to section 4.1.2 of https://dl.dropboxusercontent.com/u/12630719/cohesivedocument131024.pdf

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Urs comments on "parameterized cohesive spectra" (42510) https://nforum.ncatlab.org/discussion/5321/?Focus=42510#Comment_42510 2013-10-21T22:16:33+00:00 2023-12-09T21:33:09+00:00 Urs https://nforum.ncatlab.org/account/4/ Have now finally contacted Georg Biedermann. He kindly points out his articles where he constructs model categories of n-excisive functors, which are homotopy toposes in Charles’s terminology, ...

Have now finally contacted Georg Biedermann. He kindly points out his articles where he constructs model categories of n-excisive functors, which are homotopy toposes in Charles’s terminology, hence present the $\infty$-toposes that we are discussing.

While the Lab is unresponsive, I’ll record these reference here:

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David_Corfield comments on "parameterized cohesive spectra" (42504) https://nforum.ncatlab.org/discussion/5321/?Focus=42504#Comment_42504 2013-10-21T12:14:53+00:00 2023-12-09T21:33:09+00:00 David_Corfield https://nforum.ncatlab.org/account/20/ About those two flat tangent connections mentioned in #29, Goodwillie says in the abstract for a 2005 workshop their difference is the tensor ﬁeld known as smash product of spectra. He ...

About those two flat tangent connections mentioned in #29, Goodwillie says in the abstract for a 2005 workshop

their difference is the tensor ﬁeld known as smash product of spectra.

He continues

I will say something about higher-order jets and about differential operators. I cannot make much sense of differential forms (except 0-forms and 1-forms), but I may talk about them anyway.

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Urs comments on "parameterized cohesive spectra" (42496) https://nforum.ncatlab.org/discussion/5321/?Focus=42496#Comment_42496 2013-10-20T20:55:30+00:00 2023-12-09T21:33:10+00:00 Urs https://nforum.ncatlab.org/account/4/ I see, thanks for the information. I suppose it is about time that I contact Georg Biedermann.

I see, thanks for the information. I suppose it is about time that I contact Georg Biedermann.

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Charles Rezk comments on "parameterized cohesive spectra" (42495) https://nforum.ncatlab.org/discussion/5321/?Focus=42495#Comment_42495 2013-10-20T20:28:01+00:00 2023-12-09T21:33:10+00:00 Charles Rezk https://nforum.ncatlab.org/account/442/ Urs 51. I’m probably hallucinating. I don’t know where he would have said this, though it is true. Goodwillie does briefly discuss parameterized spectra (in Calculus 1, remark 1.6). Here he is ...

Urs 51. I’m probably hallucinating. I don’t know where he would have said this, though it is true.

Goodwillie does briefly discuss parameterized spectra (in Calculus 1, remark 1.6). Here he is thinking about functors $\mathrm{Top}/X\to \mathrm{Top}_*$ (it would be the same I think if you consider functors $X\backslash \mathrm{Top}/X\to \mathrm{Top}$), and in this case reduced $1$-excisive functors are the same as parameterized spectra over $X$.

However, this is a completely different appearence of parameterized spectra, than the one I discussed above. (Basically, an object of $\mathcal{F}_1$ is a parameterized spectrum over the space $T=F(*)$ which is the value of $F$ at the terminal object of the domain; 1-excisive functors on the slice category $X\backslash\mathrm{Top}/X$ want to be parameterized over $X$, which is itself the terminal object of the domain.)

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Charles Rezk comments on "parameterized cohesive spectra" (42493) https://nforum.ncatlab.org/discussion/5321/?Focus=42493#Comment_42493 2013-10-20T19:42:28+00:00 2023-12-09T21:33:10+00:00 Charles Rezk https://nforum.ncatlab.org/account/442/ Urs 51: I’m not sure. I think it was in Calculus 3, but that would have been a preprint version I saw many years ago. Urs 50: I told Jacob that, but I’d learned it from Biedermann.

Urs 51: I’m not sure. I think it was in Calculus 3, but that would have been a preprint version I saw many years ago.

Urs 50: I told Jacob that, but I’d learned it from Biedermann.

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Urs comments on "parameterized cohesive spectra" (42490) https://nforum.ncatlab.org/discussion/5321/?Focus=42490#Comment_42490 2013-10-20T19:10:28+00:00 2023-12-09T21:33:10+00:00 Urs https://nforum.ncatlab.org/account/4/ Goodwillie showed that […] the 1-excisive functors F equipped with an identification of P0F with the space X, is the same as parameterized spectra over X. Maybe you can save me a minute: which ...

Goodwillie showed that […] the 1-excisive functors F equipped with an identification of P0F with the space X, is the same as parameterized spectra over X.

Maybe you can save me a minute: which page is this statement on?

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Urs comments on "parameterized cohesive spectra" (42489) https://nforum.ncatlab.org/discussion/5321/?Focus=42489#Comment_42489 2013-10-20T19:04:14+00:00 2023-12-09T21:33:10+00:00 Urs https://nforum.ncatlab.org/account/4/ Oh, and in remark 7.1.1. 11 right below Lurie attributes the statement of your #47 above to you. :-) Is this in print anywhere? Do you say this in “Homotopy toposes”? (I don’t remember having ...

Oh, and in remark 7.1.1. 11 right below Lurie attributes the statement of your #47 above to you. :-)

Is this in print anywhere? Do you say this in “Homotopy toposes”? (I don’t remember having seen it there, but I may have forgotten.)

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Charles Rezk comments on "parameterized cohesive spectra" (42487) https://nforum.ncatlab.org/discussion/5321/?Focus=42487#Comment_42487 2013-10-20T18:47:58+00:00 2023-12-09T21:33:10+00:00 Charles Rezk https://nforum.ncatlab.org/account/442/ Fully explicitly, it is in theorem 7.1.1.10 of Lurie’s “Higher Algebra”. I suppose I should read that book someday.

Fully explicitly, it is in theorem 7.1.1.10 of Lurie’s “Higher Algebra”.

I suppose I should read that book someday.

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Urs comments on "parameterized cohesive spectra" (42484) https://nforum.ncatlab.org/discussion/5321/?Focus=42484#Comment_42484 2013-10-20T18:27:59+00:00 2023-12-09T21:33:10+00:00 Urs https://nforum.ncatlab.org/account/4/ Charles, thanks for pushing this further! This would be theorem 1.8 in “Calculus III”, I suppose. Fully explicitly, it is in theorem 7.1.1.10 of Lurie’s “Higher Algebra”.

Charles,

thanks for pushing this further!

This would be theorem 1.8 in “Calculus III”, I suppose.

Fully explicitly, it is in theorem 7.1.1.10 of Lurie’s “Higher Algebra”.

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Charles Rezk comments on "parameterized cohesive spectra" (42477) https://nforum.ncatlab.org/discussion/5321/?Focus=42477#Comment_42477 2013-10-20T16:22:09+00:00 2023-12-09T21:33:10+00:00 Charles Rezk https://nforum.ncatlab.org/account/442/ I understand things like this: I write Top\mathrm{Top} for the homotopy theory of spaces (i.e., &infin;\infty-groupoids). Let &Fscr;=Func filt(Top ...

I understand things like this:

I write $\mathrm{Top}$ for the homotopy theory of spaces (i.e., $\infty$-groupoids). Let $\mathcal{F}=\mathrm{Func}^{\mathrm{filt}}(\mathrm{Top}_*,\mathrm{Top})$ denote the homotopy theory of functors from pointed spaces to spaces which commute with all filtered colimits. We can identify a full subtheory $\mathcal{F}_n\subset \mathcal{F}$ of $n$-excisive functors.

Goodwillie (in “Calculus 3”) has given an explicit formula to construct the left adjoint $P_n\colon \mathcal{F}\to \mathcal{F}_n$ to the inclusion $\mathcal{F}_n\subset \mathcal{F}$, called “$n$-excisive approximation”. It’s immediate from Goodwillie’s formula that $P_n$ commutes with finite homotopy limits, and therefore each $\mathcal{F}_n$ is an $\infty$-topos. Thus we obtain a tower of $\infty$-topoi $\mathcal{F}\to \cdots \to \mathcal{F}_n\to\mathcal{F}_{n-1}\to \cdots\to \mathcal{F}_1\to \mathcal{F}_0$.

$\mathcal{F}_0$ consists of constant functors, and so is equivalent to $\mathrm{Top}$, while $\mathcal{F}_1$ is $1$-excisive functors; Goodwillie showed that $(\mathcal{F}_1)^{P_0=X}$, the $1$-excisive functors $F$ equipped with an identification of $P_0F$ with the space $X$, is the same as parameterized spectra over $X$.

“Intrinsic” characterizations of the $\mathcal{F}_n$ for $n\gt1$ seem to be much harder to get at. Probably one can say that $(\mathcal{F}_n)^{P_0=X}$ is equivalent to “parameterized reduced $n$-excisive functors over $X$”, where “reduced $n$-excisive functors” are $(\mathcal{F}_n)^{P_0=*}$. It is hard to give a characterization of $(\mathcal{F}_n)^{P_0=*}$ for $n\gt1$ that is satisfying, though Arone and Ching now have a decent answer to this, I think, in terms of comonads on spectra.

Correction. When I originally wrote this, I wrote $\mathrm{Func}^{\mathrm{filt}}(\mathrm{Top},\mathrm{Top})$ for the definition of $\mathcal{F}$, a category of functors on unbased spaces, thus perpetuating a standard confusion in this subject. There is perfectly good theory of $n$-excisive approximations for functors on unbased spaces, and it gives rise to a nice tower of $\infty$-toposes (that construction is very general)— but this tower is somewhat different than the one I actually described above. For instance, “unbased $1$-excisive functors” are the same thing as “parameterized spectra $E\to X$ equipped with a section”.

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David_Corfield comments on "parameterized cohesive spectra" (42476) https://nforum.ncatlab.org/discussion/5321/?Focus=42476#Comment_42476 2013-10-20T15:21:18+00:00 2023-12-09T21:33:10+00:00 David_Corfield https://nforum.ncatlab.org/account/20/ Urs #43, I don’t quite see how things go here. In the n=1n = 1 case, the idea is that there’s a connection between 1-excisive functors and the tangent (&infin;,1)(\infty, 1)-category ...

Urs #43, I don’t quite see how things go here. In the $n = 1$ case, the idea is that there’s a connection between 1-excisive functors and the tangent $(\infty, 1)$-category construction.

For the latter, I see how that diagram seq gets used to define parameterized spectra, and I see how that delivers the fiberwise stabilization of the codomain fibration, $Stab(Func(\Delta,C) \to C)$.

So how does this relate to 1-exciveness, which concerns taking pushout squares to pullback squares? Is the relation via the squares of seq?

If so, when you say we do the same for $k$-excisive functors, is it that seq gets replaced by a higher dimensional with $k+1$-cubes instead of squares? And what is the equivalent of $Stab(Func(\Delta,C) \to C)$? As I had it in #40?

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David_Corfield comments on "parameterized cohesive spectra" (42470) https://nforum.ncatlab.org/discussion/5321/?Focus=42470#Comment_42470 2013-10-18T11:53:37+00:00 2023-12-09T21:33:10+00:00 David_Corfield https://nforum.ncatlab.org/account/20/ With all this dash to the (&infin;,1)(\infty, 1) case, I’d forgotten Urs had already being doing tangents for (n,n)(n, n)-categories way back here and here. Did the extra generality get used, ...

With all this dash to the $(\infty, 1)$ case, I’d forgotten Urs had already being doing tangents for $(n, n)$-categories way back here and here. Did the extra generality get used, or did it mostly boil down to the $(n, 1)$ case?

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David_Corfield comments on "parameterized cohesive spectra" (42469) https://nforum.ncatlab.org/discussion/5321/?Focus=42469#Comment_42469 2013-10-18T09:51:07+00:00 2023-12-09T21:33:10+00:00 David_Corfield https://nforum.ncatlab.org/account/20/ I was thinking there should be a multivariate calculus of functors, and of course there is, see sec. 6.2. So we have a notion of n-excisive for n = (n 1,...,n m)(n_1, ..., n_m). But presumably there ...

I was thinking there should be a multivariate calculus of functors, and of course there is, see sec. 6.2. So we have a notion of n-excisive for n = $(n_1, ..., n_m)$.

But presumably there should be a multivariate jet construction. Thinking in SDG terms, we need Kock’s $D_k(n) = [(x_1, ..., x_n) \in R^n|$ product of any $k+1$ is zero].

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Urs comments on "parameterized cohesive spectra" (42464) https://nforum.ncatlab.org/discussion/5321/?Focus=42464#Comment_42464 2013-10-17T23:13:44+00:00 2023-12-09T21:33:10+00:00 Urs https://nforum.ncatlab.org/account/4/ Mike, the idea is to phrase Joyal’s observation in terms of 1-excisive functors and then observe that it goes through for kk-excisive functors, too.

Mike, the idea is to phrase Joyal’s observation in terms of 1-excisive functors and then observe that it goes through for $k$-excisive functors, too.

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